Out of idle curiosity, I'm wondering about all the various idempotent constructions we have in mathematics (they seem to be generally referred to as a "closure" or "completion"), and how some of them are related (e.g., the radical of an ideal and the closure of a subset of $k^n$ in the Zariski topology, via the Nullstellensatz - the radical and the topological closure both being idempotent).

So, one answer per post, but if you have two concepts which are related, I guess it'd be okay to put them together. For the sake of the completeness (ha ha) of this list, I'll add "radical" and "topological closure".

EDIT: My bad - I should have looked around more first. There's this list at Wikipedia and this list at nLab. Well, I'm sure there's plenty more concepts out there, so if you think of any more, feel free to add them. But let's focus on how some of these concepts are related - e.g., does one kind of completion arise in terms of another? What are some general ways in which completions and closures arise?

Yes, I'm realizing that there are some good lists already out there. I'll change the focus of the question.
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Zev ChonolesFeb 7 '10 at 17:34

Order completion is necessary to even define metric space completion since one should construct R as the order-completion of Q. So I guess one should think of metric space completion as induced from order completion.
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Qiaochu YuanFeb 7 '10 at 17:37

Well, I guess that example isn't the best, but that's the kind of thing I'm thinking of - explaining one kind of completion in terms of, or induced by, another.
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Zev ChonolesFeb 7 '10 at 17:41

1

Anyway, I think a more interesting question to ask would be something like "what are general ways in which closures and completions arise?"
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Qiaochu YuanFeb 7 '10 at 17:41

11 Answers
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In general, if $A\subset B$ is a full reflective subcategory, then each object $a\in A$ is isomorphic to its image under the reflector. This seems to include many cases: $\mathbf{Ab}\subset \mathbf{Grp}$, $\mathbf{CompMet}\subset\mathbf{Met}$, $\mathbf{Top}_{n+1}\subseteq \mathbf{Top}_{n}$ (as in Why is Top_4 a reflective subcategory of Top_3?), etc.

EDIT: Following Pete L. Clark's comment, here is a clarification: The subcategory $A$ above is called reflective if the inclusion functor $A\subset B$ has a left adjoint, and full if this inclusion functor is full. In case $A$ is a reflective subcategory, the left adjoint to the inclusion functor is called a reflector.

This is the point, I think. (Though it might be worth restating it in more broadly familiar categorical language: e.g. adjoint to the forgetful functor.) The listing of examples ad infinitum is not as insightful or useful as this one observation.
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Pete L. ClarkFeb 7 '10 at 22:22

@PLC: Thank you very much for your comment. I have edited the response accordingly for clarification.
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user2734Feb 7 '10 at 22:43

The coolest closure operation I know occurs in Razborov's lower bound for the monotone circuit complexity of the clique function. In that proof he needs a class of "simple" set systems, and to get it he defines a very ingenious k-ary operation on sets and defines a simple set system to be one that is closed under that operation. I don't know what general moral to draw from that, but it's fairly different from the examples mentioned so far.

Just about every form of compactification. The compactification of a compact space is itself, and a compactification had better be compact or it shouldn't be called a compactification.

The same thing goes for completion of metric spaces, of course. (I know, shouldn't post two answers in one for this kind of list, but they're closely related and trivially known for everybody. I put them here for completeness' sake (no pun intended).)

Ah! You edited your question! I had to delete my answer. Anyway, here is a general scenario where idempotent operations such as the one you want arise:

In this para I am going to be vague. But the examples below given should illustrate what I have in mind.Ok, so, You have "some structure" somewhere. You want to go to the "maximal" of such a thing. You have a natural ordering on such structures you want. And also it so happens that the union of a chain of such stuff is again such a thing. Then you apply Zorn's lemma to find the maximal thing. And this operation of going and finding the maximal thing is an "idempotent completion" in your sense.

There are plenty of examples. A few:

$1$. A set of linearly independent vectors in a vector space is enlarged to a basis.

$2$. An algebraic extension of a field is enlarged to the algebraic closure.

$3$. A separable extension of a field, is enlarged to separable closure.

$4$. A differentiable atlas on a smooth manifold is enlarged to a maximal one, ie., a differentiable structure.

$5$. A certain functional on a Banach space is enlarged to fill the whole space, as in the proof of Hahn-Banach theorem.

And so on, nearly in fact every application of Zorn's lemma.

This is not a functorial way to go; but the construction as an operation is idempotent. And in some way, such as in the construction of the algebraic closure, we have an isomorphism of two such different constructions.

In Model theory we have Skolem hulls:
Assuming that each formula has a corresponding Skolem function, one can take a given subset A of a model M, and close it under Skolem functions. This closure Sk(A) is the smallest elementary submodel of M containing A.